Given $z$ is a complex number such that $Arg(z) = theta$ and $|z| = r$ , write an expression for each of $Arg(4z^3)$ and $|4z^3|$ in terms of r and/or θ. How should I approach this?

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Answer 1 · 2018-06-09T06:53:16

Contd.........

A complex no. $z$ is such that $Arg(z)=theta and |z|=r$ .

Clearly, $z=r(costheta+isintheta)$ .

By De Moivre's Theorem, $z^n=r^n(cosntheta+isinntheta)$ .

$:. z^3=r^3(cos3theta+isin3theta)$ .

$:. 4z^3=4r^3(cos3theta+isin3theta)$ .

Answer 2 · 2018-06-09T07:06:52

$z = r e^(i theta)$

$4 z^3 = 4(r e^(i theta))^3 = 4 r^3 e^(i 3 theta)$

Given zz is a complex number such that Arg(z) = thetaArg(z) = theta and |z| = r|z| = r , write an expression for each of Arg(4z^3)Arg(4z^3) and |4z^3||4z^3| in terms of r and/or θ. How should I approach this?